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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{复变函数练习6.1-6.2 - 留数、用留数定理计算实积分 }
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\date{2024 年 6 月 3 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
（柯西留数定理）设 $f(z)$ 在周线或复周线 $C$ 所围的区域 $D$ 内，除 $a_1,a_2,\cdots,a_n$ 外解析，
在闭域 $\overline{D}=D\cup C$ 上除 $a_1,a_2,\cdots,a_n$ 外连续，则
$$\int_Cf(z)dz = 2\pi i \sum\limits_{k=1}^{n} {\mathrm{Res}}_{z=a_k} f(z). $$

\vspace{0.2cm}

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\item  %Problem 02
（例子6.1）计算积分 $$\int_{|z|=2} \frac{5z-2}{z(z-1)^2}dz. $$

\vspace{0.2cm}

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\item  %Problem 03
（例子6.2）计算积分 $$\int_{|z|=n} \tan \pi z dz. $$


\vspace{0.2cm}

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\item  %Problem 04
（例子6.3）计算积分 $$\int_{|z|=1} \frac{\cos z}{z^3}dz. $$


\vspace{0.2cm}

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\item  %Problem 05
（例子6.4）计算积分 $$\int_{|z|=1} \frac{z\sin z}{(1-e^z)^3}dz. $$


\vspace{0.2cm}

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\item  %Problem 06
（例子6.7）设 $0\le |p|<1$, 计算积分 $$\int_0^{2\pi} \frac{d\theta}{1-2p\cos\theta+p^2}. $$


\vspace{0.2cm}

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\item  %Problem 07
（例子6.8）设 $a>b>0$, 计算积分 $$\int_0^{2\pi} \frac{\sin^2\theta d\theta}{a+b\cos\theta}. $$


\vspace{0.2cm}


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\item  %Problem 08
（例子6.11）设 $a>0$, 计算积分 $$\int_0^{+\infty} \frac{dx}{x^4+a^4}. $$


\vspace{0.2cm}

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\item  %Problem 09
（例子6.13）设 $m>0$, 计算积分 $$\int_0^{+\infty} \frac{\cos mx}{1+x^2}dx. $$


\vspace{0.2cm}

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\item  %Problem 10
（例子6.15）计算积分 $$\int_0^{+\infty} \frac{\sin x}{x}dx. $$


\vspace{0.2cm}


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\end{enumerate}


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\end{document}

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